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A250207 The number of quartic terms in the multiplicative group modulo n. 6
1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 3, 1, 1, 4, 3, 9, 1, 3, 5, 11, 1, 5, 3, 9, 3, 7, 1, 15, 2, 5, 4, 3, 3, 9, 9, 3, 1, 10, 3, 21, 5, 3, 11, 23, 1, 21, 5, 4, 3, 13, 9, 5, 3, 9, 7, 29, 1, 15, 15, 9, 4, 3, 5, 33, 4, 11, 3, 35, 3, 18, 9, 5, 9, 15, 3, 39, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

In the character table of the multiplicative group modulo n there are phi(n) different characters. [This is made explicit for example by the number of rows in arXiv:1008.2547.] The set of the fourth powers of the characters in all representations has some cardinality, which defines the sequence.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

R. J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, (2017).

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010.

Wikipedia, Dirichlet character

FORMULA

a(n) = A000010(n)/A073103(n).

Multiplicative with a(2^e) = 1 for e<=3; a(2^e) = 2^(e-4) for e>=4; a(p^e) = p^(e-1)*(p-1)/4 for e>=1 and p == 1 (mod 4); a(p^e) = p^(e-1)*(p-1)/2 for e>=1 and p == 3 (mod 4). (Derived from A073103.) - R. J. Mathar, Oct 13 2017

EXAMPLE

For n <= 6, the set of all characters in all representations consists of a subset of +1, -1, +i or -i. Their fourth powers are all +1, a single value, so a(n)=1 then.

For n=7, the set of characters is 1, -1, +-1/2 +- sqrt(3)*i/2, so their fourth powers are 1 or -1/2 +- sqrt(3)*i/2, which are three different values, so a(7)=3.

For n=11, the fourth powers of the characters may be 1, exp(+-2*i*Pi/5) or exp(+-4*i*Pi/5), which are 5 different values.

MAPLE

A250207 := proc(n)

    numtheory[phi](n)/A073103(n) ;

end proc:

MATHEMATICA

a[n_] := EulerPhi[n]/Count[Range[0, n-1]^4 - 1, k_ /; Divisible[k, n]];

Array[a, 80] (* Jean-François Alcover, Nov 20 2017 *)

PROG

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]==2, 2^max(0, f[i, 2]-4), f[i, 1]^(f[i, 2]-1)*(f[i, 1]-1)/if(f[i, 1]%4==1, 4, 2))) \\ Charles R Greathouse IV, Mar 02 2015

CROSSREFS

Cf. A046073, A087692, A052273, A293482 - A293485.

Sequence in context: A213621 A053575 A293485 * A216319 A309425 A218355

Adjacent sequences:  A250204 A250205 A250206 * A250208 A250209 A250210

KEYWORD

easy,nonn,mult

AUTHOR

R. J. Mathar, Mar 02 2015

STATUS

approved

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Last modified November 14 15:10 EST 2019. Contains 329126 sequences. (Running on oeis4.)